Biomedical Engineering Reference
In-Depth Information
is assumed to be aligned with the
x
axis after collision,
v
(
t
)
=
v
x
0
x
,
andiftherewerenoexternalforcespresent,themeancollisiontime,
the time taken for the particle to travel the mean free path distance
d
would be
t
col
v
x
0
. The modified time taken between collisions,
t
m
,
for an ion moving under the influence of a uniform static E field
alignedwiththe
x
axis,canbeobtainedanalyticallyassuming
t
m
and
t
col
<<
1.
q
2
m
E
0
v
x
0
t
m
+
t
m
−
t
col
=
0
(4.11a)
and the changein position can bedescribed as
1
2
qE
x
m
t
col
x
=
v
x
0
t
col
+
(4.11b)
For the case of mobility due to a 1 V m
−
1
E field, the relative
change in time over a single step is equal to the ratio of mean
velocity offset to mean thermal velocity, 3.1
10
−
10
, ratios similar
tothosefoundbyBarnes(1986)andCiccottiandJacucci(1975).The
much larger noise fields are not able to drown out the ion mobility
where enough time or su
cient field levels exist. Likewise, the
positional offset during one collision can be estimated analytically
to be a tiny 4.0
×
10
−
22
m. The random walk model was checked
using a single step of the walk with a 1 V m
−
1
Efieldinthe
X
direction. The positional offset was numerically calculated to be
3.92
×
10
−
22
m, comparing well with the analytic result. Note that
this positional offset is tiny even compared with the dimensions
of the ion (10
−
10
m). We picture a very crowded collision process
that takes many collisions to move substantial distances due to the
accumulation of offsets of a 1 V m
−
1
E field. Such tiny distances are
not unique to the random walk model but are also associated with
the viscous model; experimental mobility rates also indicate that
positionaldrift isvery tinywhen taken over onecollisionperiod.
It can be seen from Fig. 4.16a,b, in regard to ensemble effects,
thatEfieldoffsetsaredirectedalongtheappliedfield.Forauniform
spatial distribution of ionic particles, as exists under equilibrium
conditions, the offsets from individual ions sum together to affect
equilibrium.Thesameappliestonon-equilibriumconditions.These
effects are related to the ability of the E field to do work as motion
proceeds.
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